View source for IBM Model 2

== Model ==

One of the problems of the [[IBM Model 1]] is that it is very weak to reordering, since <math>p(f,a|s)</math> is calculated using only the lexical translation probabilities <math>tr(t|s)</math>. Because of this, if the model is presented with 2 translations candidates <math>t_1</math> and <math>t_2</math> with the same lexical translations, but with different reordering of the translated words, the model scores both translations with the same score. 

Mixture-based Alignment models~(IBM Model 2) addresses this problem by modeling the absolute distortion in the word positioning between the 2 languages, introducing an alignment probability distribution <math>Pr_a(i|j,J,I)</math>, where <math>i</math> and <math>j</math> are the word positions in the source and target sentences. Thus the equation for <math>Pr(t,a|s)</math> becomes:

<math>
Pr(t,a|s) = \frac{\epsilon}{(J+1)^{I}}\prod_{j=1}^{J}{tr(t_j|s_{a(j)}) Pr_a(a(j)|j,J,I)}
</math>

Where the alignment probability distribution <math>Pr_a(a(j)|j,J,I)</math> models the probability of a word in the position <math>i</math> in the source sentence of being reordered into the position <math>j</math> in the target sentence.